Rayo's number is a large number named after Mexican philosophy professor Agustín Rayo which has been claimed to be the largest named number.[1][2] It was originally defined in a "big number duel" at MIT on 26 January 2007.[3][4]

Definition edit

The definition of Rayo's number is a variation on the definition:[5]

The smallest number bigger than any finite number named by an expression in the language of first-order set theory with a googol symbols or less.

Specifically, an initial version of the definition, which was later clarified, read "The smallest number bigger than any number that can be named by an expression in the language of first-order set-theory with less than a googol ( ) symbols."[4]

The formal definition of the number uses the following second-order formula, where   is a Gödel-coded formula and   is a variable assignment:[5]


Given this formula, Rayo's number is defined as:[5]

The smallest number bigger than every finite number   with the following property: there is a formula   in the language of first-order set-theory (as presented in the definition of  ) with less than a googol symbols and   as its only free variable such that: (a) there is a variable assignment   assigning   to   such that  , and (b) for any variable assignment  , if  , then   assigns   to  .

Explanation edit

Intuitively, Rayo's number is defined in a formal language, such that:

  •   and   are atomic formulas.
  • If   is a formula, then   is a formula (the negation of  ).
  • If   and   are formulas, then   is a formula (the conjunction of   and  ).
  • If   is a formula, then   is a formula (existential quantification).

Notice that it is not allowed to eliminate parentheses. For instance, one must write   instead of  .

It is possible to express the missing logical connectives in this language. For instance:

  • Disjunction:   as  .
  • Implication:   as  .
  • Biconditional:   as  .
  • Universal quantification:   as  .

The definition concerns formulas in this language that have only one free variable, specifically  . If a formula with length   is satisfied iff   is equal to the finite von Neumann ordinal  , we say such a formula is a "Rayo string" for  , and that   is "Rayo-nameable" in   symbols. Then,   is defined as the smallest   greater than all numbers Rayo-nameable in at most   symbols.

Examples edit

To Rayo-name  , which is the empty set, one can write  , which has 10 symbols. It can be shown that this is the optimal Rayo string for  .[citation needed] Similarly,  , which has 30 symbols, is the optimal string for  .[citation needed] Therefore,   for  , and   for  .

Additionally, it can be shown that   and   (tetration).[citation needed]

References edit

  1. ^ "CH. Rayo's Number". The Math Factor Podcast. Retrieved 24 March 2014.
  2. ^ Kerr, Josh (7 December 2013). "Name the biggest number contest". Archived from the original on 20 March 2016. Retrieved 27 March 2014.
  3. ^ Elga, Adam. "Large Number Championship" (PDF). Archived from the original (PDF) on 14 July 2019. Retrieved 24 March 2014.
  4. ^ a b Manzari, Mandana; Nick Semenkovich (31 January 2007). "Profs Duke It Out in Big Number Duel". The Tech. Retrieved 24 March 2014.
  5. ^ a b c Rayo, Agustín. "Big Number Duel". Retrieved 24 March 2014.